Question: Solve for $x$ : $ 4|x + 6| - 6 = -6|x + 6| + 4 $
Answer: Add $ {6|x + 6|} $ to both sides: $ \begin{eqnarray} 4|x + 6| - 6 &=& -6|x + 6| + 4 \\ \\ { + 6|x + 6|} && { + 6|x + 6|} \\ \\ 10|x + 6| - 6 &=& 4 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 10|x + 6| - 6 &=& 4 \\ \\ { + 6} &=& { + 6} \\ \\ 10|x + 6| &=& 10 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x + 6|} {{10}} = \dfrac{10} {{10}} $ Simplify: $ |x + 6| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 6 = -1 $ or $ x + 6 = 1 $ Solve for the solution where $x + 6$ is negative: $ x + 6 = -1 $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& -1 \\ \\ {- 6} && {- 6} \\ \\ x &=& -1 - 6 \end{eqnarray} $ $ x = -7 $ Then calculate the solution where $x + 6$ is positive: $ x + 6 = 1 $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& 1 \\ \\ {- 6} && {- 6} \\ \\ x &=& 1 - 6 \end{eqnarray} $ $ x = -5 $ Thus, the correct answer is $x = -7 $ or $x = -5 $.